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I want to comment on your last question: the classes. The theorems mentioned rightly belong to the area of algebraic topology (the deep ideas behind them are of topological nature and are usually exp …

This is inspired by The Whitehead for maps question. Consider two maps f, g: X\to Y which happen to induce the same maps (of discrete spaces) [Z, X] \to [Z, Y] for every Z. Does this mean f and g are …

The post consists of initial ideas on top and the proof at the bottom. I think the key idea is to perform the process opposite to what you describe about subdividing into triangles. Indeed, our pro …

I found it myself: the image of id \in [X, X] under both maps will be the same class in [X, Y], which is the definition of homotopy between f and g, so the ansewr is yes.

This question has been inspired by covering 3-torus post. Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away …

Kind of late to the party, but the (weak) contractibility follows from $\pi_i(S^\infty) = 0$ for $i>0$.

What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?

I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology the …

To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is r …